Data Structures – Week #1 Introduction

February 17, 2012Borahan Tümer, Ph.D.2 Goals We will learn methods of how to (explicit goal) organize or structure large amounts of data in the main memory (MM) considering efficiency; i.e, memory space and execution time (implicit goal) gain additional experience on what data structures to use for solving what kind of problems programming

February 17, 2012Borahan Tümer, Ph.D.3 Goals continued…1 Explicit Goal We look for answers to the following question: “How do we store data in MM such that execution time grows as slow as possible with the growing size of input data, and data uses up minimum memory space?”

February 17, 2012Borahan Tümer, Ph.D.4 Goals continued…2 As a tool to calculate the execution time of algorithms, we will learn the basic principles of algorithm analysis. To efficiently structure data in MM, we will thoroughly discuss the static, (arrays) dynamic (pointers)‏ ways of memory allocations, two fundemantal Implementation tools for data structures.

February 17, 2012Borahan Tümer, Ph.D.5 Representation of Main Memory

February 17, 2012Borahan Tümer, Ph.D.6 Examples for efficient vs. inefficient data structures 8-Queen problem 1D array vs. 2D array representation results in saving memory space Search for proper spot (square) using horse moves save time over square-by-square search Fibonacci series: A lookup table avoids redundant recursive calls and saves time

February 17, 2012Borahan Tümer, Ph.D.7 Examples for efficient vs. inefficient data structures 8-Queen problem x x x x x x x x x x x x x x x x x x

February 17, 2012Borahan Tümer, Ph.D.8 Examples for efficient vs. inefficient data structures 8-Queen problem x x x x x x x x x int a[4][4]; …. a[0][1]=1; a[1][3]=1; a[2][0]=1; a[3][2]=1; int a[5]; …. a[0]=0; a[1]=2; a[2]=4; a[3]=1; a[4]=3; inefficient: more memory space required efficient: less memory space required

February 17, 2012Borahan Tümer, Ph.D.9 Math Review Exponents Logarithms

February 17, 2012Borahan Tümer, Ph.D.10 Math Review Arithmetic Series: Series where the variable of summation is the base.

February 17, 2012Borahan Tümer, Ph.D.11 Math Review Geometric Series: Series at which the variable of summation is the exponent.

February 17, 2012Borahan Tümer, Ph.D.12 Math Review Geometric Series…cont’d An example to using above formulas to calculate another geometric series

February 17, 2012Borahan Tümer, Ph.D.13 Math Review Proofs Proof by Induction Steps – Prove the base case (k=1)‏ – Assume hypothesis holds for k=n – Prove hypothesis for k=n+1 Proof by counterexample Prove the hypothesis wrong by an example Proof by contradiction ( )‏ Assume hypothesis is wrong, Try to prove this See the contradictory result

February 17, 2012Borahan Tümer, Ph.D.14 Math Review Proof examples (Proofs… cont’d)‏ Proof by Induction Hypothesis Steps 1. Prove true for n=1: 2. Assume true for n=k: 3. Prove true for n=k+1:

February 17, 2012Borahan Tümer, Ph.D.15 Arrays Static data structures that represent contiguous memory locations holding data of same type provide direct access to data they hold have a constant size determined up front (at the beginning of) the run time

February 17, 2012Borahan Tümer, Ph.D.16 Arrays… cont’d An integer array example in C int arr[12]; //12 integers

February 17, 2012Borahan Tümer, Ph.D.17 Multidimensional Arrays To represent data with multiple dimensions, multidimensional array may be employed. Multidimensional arrays are structures specified with the data value, and as many indices as the dimensions of array Example: int arr2D[r][c];

February 17, 2012Borahan Tümer, Ph.D.18 Multidimensional Arrays m: a two dimensional (2D) array with r rows and c columns Row-major representation: 2D array is implemented row-by-row. Column-major representation: 2D array is implemented column-first. In row-major rep., m[i][j] is the entry of the above matrix m at i+1 th row and j+1 th column. “i” and “j” are row and column indices, respectively. How many elements? n = r*c elements

February 17, 2012Borahan Tümer, Ph.D.19 Row-major Implementation Question: How can we store the matrix in a 1D array in a row-major fashion or how can we map the 2D array m to a 1D array a?...m[0][0]...m[0][c-1]...m[r-1][0]... m[r-1][c-1] a l elements index: k → k=l k=l+c-1... k=l+(r-1)c+0k=l+(r-1)c+c-1 In general, m[i][j] is placed at a[k] where k=l+ic+j.

February 17, 2012Borahan Tümer, Ph.D.20 Implementation Details of Arrays Array names are pointers that point to the first byte of the first element of the array. a) double vect[row_limit];// vect is a pointer!!! Arrays may be efficiently passed to functions using their name and their size where the name specifies the beginning address of the array the size states the bounds of the index values. 1. Arrays can only be copied element by element.

February 17, 2012Borahan Tümer, Ph.D.21 Implementation Details… cont’d #define maxrow …; #define maxcol …; … int main() { int minirow; double min; double probability_matrix[maxrow][maxcol]; … ; //probability matrix initialized!!! min=minrow(probability_matrix,maxrow,maxcol,&minirow); …; return 0; }

February 17, 2012Borahan Tümer, Ph.D.22 Implementation Details… cont’d double minrow(double arr[][maxcol], int xpos, int ypos, int *ind){ // finds minimum of sum of rows of the matrix and returns the sum // and the row index with minimum sum. double min; …; mn = ; for (i=0; i<=xpos; i++) { sum=0; for (j=0; j<=ypos; j++)‏ sum += darr[i][j]; if (mn > sum) { mn=sum; *ind=i; } // call by reference!!! } return mn; }

February 17, 2012Borahan Tümer, Ph.D.23 Records As opposed to arrays in which we keep data of the same type, we keep related data of various types in a record. Records are used to encapsulate (keep together) related data. Records are composite, and hence, user-defined data types. In C, records are formed using the reserved word “ struct.”

February 17, 2012Borahan Tümer, Ph.D.24 Struct We declare as an example a student record called “ stdType ”. We declare first the data types required for individual fields of the record stdType, and then the record stdType itself.

February 17, 2012Borahan Tümer, Ph.D.25 Struct… Example enum genderType = {female, male}; // enumerated type declared… typedef enum genderType genderType; // name of enumerated type shortened… struct instrType { …// left for you as exercise!!! } typedef struct instrType instrType; struct classType { // fields (attributes in OOP) of a course declared… char classCode[8]; char className[60]; instrType instructor; struct classType *clsptr; } typedef struct classType classType; // name of structure shortened…

February 17, 2012Borahan Tümer, Ph.D.26 Struct… Example continues struct stdType { char id[8]; //key //personal info char name[15]; char surname[25]; genderType gender;//enumerated type … //student info classType current_classes[10]; //...or classType *cur_clsptr classType classes_taken[50]; //...or classType *taken_clsptr float grade; unsigned int credits_earned; … //next record’s first byte’s address struct stdType *sptr; //address of next student record }

February 17, 2012Borahan Tümer, Ph.D.27 Memory Issues Arrays can be used within records. Ex: classType current_classes[10]; // from previous slide Each element of an array can be a record. stdType students[1000]; Using an array of classType for keeping taken classes wastes memory space (Why?)‏ Any alternatives? How will we keep student records in MM? In an array? Advantages? Disadvantages?

February 17, 2012Borahan Tümer, Ph.D.28 Array Representation Advantages Direct access (i.e., faster execution)‏ Disadvantages Not suitable for changing number of student records The higher the extent of memory waste the smaller the number of student records required to store than that at the initial case. The (constant) size of array requires extension which is impossible for static arrays in case the number exceeds the bounds of the array. The other alternative is pointers that provide dynamic memory allocation

February 17, 2012Borahan Tümer, Ph.D.29 Pointers Pointers are variables that hold memory addresses. Declaration of a pointer is based on the type of data of which the pointer holds the memory address. Ex: stdtype *stdptr;

February 17, 2012Borahan Tümer, Ph.D.30 Linked List Representation Value of header=?Value of header=2E450 Std #1Std #2Std #3Std #n header Memory heap Std 1 Std 2 Std 3 Std n header

February 17, 2012Borahan Tümer, Ph.D.31 Dynamic Memory Allocation header=(*stdtype) malloc(sizeof(stdtype)); //Copy the info of first student to node pointed to by header s =(*stdtype) malloc(sizeof(stdtype)); //Copy info of second student to node pointed to by header Header->sptr=s;...

February 17, 2012Borahan Tümer, Ph.D.32 Arrays vs. Pointers Static data structures Represented by an index and associated value Consecutive memory cells Direct access (+)‏ Constant size (-)‏ Memory not released during runtime (-)‏ Dynamic data structures Represented by a record of information and address of next node Randomly located in heap (cause for need to keep address of next node)‏ Sequential access (-)‏ Flexible size (+)‏ Memory space allocatable and releasable during runtime (+)‏